# zbMATH — the first resource for mathematics

A charge simulation method for the numerical conformal mapping of interior, exterior and doubly-connected domains. (English) Zbl 0818.30004
It is well known that the conformal mapping of a simply [or doubly] connected domain $$D$$ onto the unit disk [or a circular ring] can be reduced to the solution of a Dirichlet problem. The author solves the latter by the ‘charge simulation method’ which has been studied extensively by several Japanese authors: Murashima, Katsurada, Okamoto and the present author. In this method the solution of the Dirichlet problem is approximated by $$G(z)= \sum^ N_{i= 1} Q_ i\log | z- \zeta_ i|$$ with unknown coefficients $$Q_ i$$, where the charge points $$\zeta_ i$$ lie outside $$\overline D$$. To determine the $$Q_ i$$ it is required that $$G(z_ j)= b(z_ j)$$ for $$N$$ collocation points $$z_ j\in \partial D$$. The error is defined by $$E_ G= \max| G(z_{j+{1\over 2}})- b(z_{j+{1\over 2}})|$$, where $$z_{j+{1\over 2}}\in \partial D$$ are intermediate points. The determination of $$G$$ thus leads to the solution of a $$N\times N$$ linear system for the $$Q_ i$$. – The paper fully covers earlier work and gives a unified treatment of three types of mapping problems. The choice of the charge points is discussed, and the condition of the $$N\times N$$ matrix is studied. Results of 11 numerical examples are given and compared with previous experiments. Good results are reported in all cases provided that no concave corners on $$\partial D$$ occur.
Reviewer: D.Gaier (Gießen)

##### MSC:
 30C30 Schwarz-Christoffel-type mappings
##### Keywords:
charge simulation method
Full Text:
##### References:
 [1] Amano, K., Numerical conformal mapping based on the charge simulation method, Trans. inform. process. soc. Japan, 28, 697-704, (1987), (in Japanese) [2] Amano, K., Numerical conformal mapping of exterior domains based on the charge simulation method, Trans. inform. process. soc. Japan, 29, 62-72, (1988), (in Japanese) [3] Amano, K., Numerical conformal mapping of doubly-connected domains by the charge simulation method, Trans. inform. process. soc. Japan, 29, 914-924, (1988), (in Japanese) [4] Amano, K., A bidirectional method for numerical conformal mapping based on the charge simulation method, J. inform. process., 14, 473-482, (1991) · Zbl 0769.30003 [5] Berrut, J.-P.; Berrut, J.-P., A Fredholm integral equation of the second kind for conformal mapping, (), 14, 1&2, 99-110, (1986), also: in · Zbl 0577.30010 [6] Fairweather, G.; Johnston, R.L., The method of fundamental solutions for problems in potential theory, (), 349-359 · Zbl 0546.76021 [7] Gaier, D., Konstruktive methoden der konformen abbildung, (1964), Springer Berlin, in German · Zbl 0132.36702 [8] Gaier, D., Integralgleichungen erster art und konforme abbildung, Math. Z., 147, 113-129, (1976), (in German) · Zbl 0304.30006 [9] Gaier, D., Das logarithmische potential und die konforme abbildung mehrfach zusammenhängender gebiete, (), 290-303, in German [10] Hayes, J.K.; Kahaner, D.K.; Kellner, R.G., An improved method for numerical conformal mapping, Math. comp., 26, 327-334, (1972) · Zbl 0239.65033 [11] Henrici, P., Fast Fourier methods in computational complex analysis, SIAM rev., 21, 4, 481-527, (1979) · Zbl 0416.65022 [12] Henrici, P., Applied and computational complex analysis, 3, (1986), Wiley New York [13] Hough, D.M.; Papamichael, N., The use of splines and singular functions in an integral equation method for c conformal mapping, Numer. math., 37, 133-147, (1981) · Zbl 0441.30016 [14] Hough, D.M.; Papamichael, N., An integral equation method for the numerical conformal mapping of interior, exterior and doubly-connected domains, Numer. math., 41, 287-307, (1983) · Zbl 0489.30008 [15] Katsurada, M., A mathematical study of the charge simulation method II, J. fac. sci. univ. Tokyo sect. IA math., 36, 135-162, (1989) · Zbl 0681.65081 [16] Katsurada, M., Asymptotic error analysis of the charge simulation method in a Jordan region with an analytic boundary, J. fac. sci. univ. Tokyo sect. IA math., 37, 635-657, (1990) · Zbl 0723.65093 [17] Katsurada, M.; Okamoto, H., A mathematical study of the charge simulation method I, J. fac. sci. univ. Tokyo sect. IA math., 35, 507-518, (1988) · Zbl 0662.65100 [18] Kitagawa, T., On the numerical stability of the method of fundamental solution applied to the Dirichlet problem, Japan J. appl. math., 5, 123-133, (1988) · Zbl 0644.65060 [19] Mathon, R.; Johnston, R.L., The approximate solution of elliptic boundary-value problems by fundamental solutions, SIAM J. numer. anal., 14, 638-650, (1977) · Zbl 0368.65058 [20] Murashima, S., Charge simulation method and its application, (1983), Morikita Tokyo, in Japanese · Zbl 0541.73098 [21] Murashima, S.; Kuhara, H., An approximate method to solve two-dimensional Laplace’s equation by means of superposition of Green’s functions on a Riemann surface, J. inform. process., 3, 127-139, (1980) · Zbl 0444.65079 [22] Reichel, L.; Reichel, L., A fast method for solving certain integral equations of the first kind with application to conformal mapping, (), 14, 1&2, 125-142, (1986), also: in: · Zbl 0587.30007 [23] Singer, H.; Steinbigler, H.; Weiss, P., A charge simulation method for the calculation of high voltage fields, IEEE trans. power apparatus systems, 1660-1668, (1974), PAS-93 [24] Steinbigler, H., Anfangsfeldstärken und ausnutzungsfaktoren rotationssymmetrischer elektrodenanordnungen in luft, Doctoral thesis, (1969), Technische Hochschule Munich [25] Symm, G.T., An integral equation method in conformal mapping, Numer. math., 9, 250-258, (1966) · Zbl 0156.16901 [26] Symm, G.T., Numerical mapping of exterior domains, Numer. math., 10, 437-445, (1967) · Zbl 0155.21502 [27] Symm, G.T., Conformal mapping of doubly-connected domains, Numer. math., 13, 448-457, (1969) · Zbl 0174.20602 [28] ()
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.